Let F(x) E C'[0, 1] and F(x) be twice differentiable in (0, 1). Suppose that f(x) = F'(x) E R[0, 1] and that |F"(x)| < M for all x E (0, 1). Show that for some constant K > 0, 2 - 1 || f(æ)dx – f( < Kn-1 i=1 Hint: apply Taylor expansion with the Lagrange Remainder on each interval [(i – 1)/n, i/n]. Remark 1: This gives the speed of convergence of the left Riemann sum. Remark 2: We will learn that continuous functions are always Riemann-integrable so the assumption f E R[0, 1] is unnecessary.
Let F(x) E C'[0, 1] and F(x) be twice differentiable in (0, 1). Suppose that f(x) = F'(x) E R[0, 1] and that |F"(x)| < M for all x E (0, 1). Show that for some constant K > 0, 2 - 1 || f(æ)dx – f( < Kn-1 i=1 Hint: apply Taylor expansion with the Lagrange Remainder on each interval [(i – 1)/n, i/n]. Remark 1: This gives the speed of convergence of the left Riemann sum. Remark 2: We will learn that continuous functions are always Riemann-integrable so the assumption f E R[0, 1] is unnecessary.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Let F(x) E C'[0, 1] and F(x) be twice differentiable in (0, 1). Suppose that f(x) = F'(x) E R[0, 1] and that
|F"(x)| < M for all x E (0,1). Show that for some constant K > 0,
n
< Kn-1.
Hint: apply Taylor expansion with the Lagrange Remainder on each interval [(i – 1)/n, i/n].
Remark 1: This gives the speed of convergence of the left Riemann sum.
Remark 2: We will learn that continuous functions are always Riemann-integrable so the assumption f E R[0, 1] is
unnecessary.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F12412377-2c97-46fd-ad41-99f2838d9383%2Fc5a803aa-e0af-44c9-8867-0d69eb5c7576%2Fx8bftjf_processed.png&w=3840&q=75)
Transcribed Image Text:Let F(x) E C'[0, 1] and F(x) be twice differentiable in (0, 1). Suppose that f(x) = F'(x) E R[0, 1] and that
|F"(x)| < M for all x E (0,1). Show that for some constant K > 0,
n
< Kn-1.
Hint: apply Taylor expansion with the Lagrange Remainder on each interval [(i – 1)/n, i/n].
Remark 1: This gives the speed of convergence of the left Riemann sum.
Remark 2: We will learn that continuous functions are always Riemann-integrable so the assumption f E R[0, 1] is
unnecessary.
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